Continued Gravitational Collapse for Newtonian Stars

The classical model of an isolated selfrgavitating gaseous star is given by the Euler-Poisson system with a polytropic pressure law $P(\rho)=\rho^\gamma$, $\gamma>1$. For any $1<\gamma<\frac43$, we construct an infinite-dimensional family of collapsing solutions to the Euler-Poisson system whose density is in general space inhomogeneous and undergoes gravitational blowup along a prescribed space-time surface, with continuous mass absorption at the origin. The leading order singular behavior is described by an explicit collapsing solution of the pressureless Euler-Poisson system.

Here star is modelled as a compactly supported compressible gas surrounded by vacuum, which interacts with a self-induced gravitational field. To describe the motion of the boundary of the star we must consider the corresponding free-boundary formulation of (1.1). In this case, a further unknown in the problem is the support of ρ(t, ·) denoted by Ω(t). We prescribe the natural boundary conditions on ∂Ω(t), (1.3a) V(∂Ω(t)) = u · n on ∂Ω(t), (1.3b) and the initial conditions (ρ(0, ·), u(0, ·)) = (ρ 0 , u 0 ) , Ω(0) = Ω. (1.4) Here V(∂Ω(t)) is the normal velocity of the moving boundary ∂Ω(t) and condition (1.3b) simply states that the movement of the boundary in normal direction is determined by the normal component of the velocity vector field. We refer to the system (1.1)-(1.3) as the EP γ -system. We point the reader to the classical text [10] where the existence of static solutions of EP γ is studied under the natural boundary condition (1.3a). We next impose the physical vacuum condition on the initial data: Condition (1.5) implies that the normal derivative of the squared speed of sound c 2 s (ρ) = dP dρ (ρ) is discontinuous at the vacuum boundary. This condition is famously satisfied by the well-known class of steady states of the EP γ -system known as the Lane-Emden stars. At the same time, condition (1.5) is the key assumption that guarantees the well-posedness of the Euler-Poisson system with vacuum regions.
Observe that forε ≪ 1 the factor ε in front of the pressure in (1.7b) is small precisely in the supercritical range 1 < γ < 4 3 . The system obtained by dropping the ε-term in (1.7b) is known as the pressureless-or dust-Euler system. This gives a vague heuristics that, if one for a moment thinks of ε as a sufficiently small length scale of density concentration, the effects of the pressure term may become negligible and the leading order singular behavior will be driven by the pressure-less dynamics. On the other hand, at this stage, this scaling heuristics is at best doubtful, as the pressure term enters the equation at the top order from the point of view of the derivative count.

Lagrangian coordinates
To address the problem of collapse we express (1.8) in the Lagrangian coordinates. Firstly, if we wish to follow the collapse process in its entirety until all of the stellar mass is absorbed, it is clear that the Eulerian description becomes inadequate at and after the first collapse time. In order to describe particle trajectories after the first collapse time, it is advantageous to work in a coordinate system that avoids this issue. Secondly, the free boundary is automatically fixed in Lagrangian description and thus more amenable to rigorous analysis. For the remainder of the paper we make the assumption of radial symmetry and assume that the reference domainΩ is the unit ball {y ∈ R 3 |y| ≤ 1}. Let the flow map η :Ω →Ω(s) be a solution of: ∂ s η(s, y) =ũ(s, η(s, y)), (1.10) η(0, y) = η 0 (y). (1.11) Here the boundary ∂Ω is mapped to the moving boundary ∂Ω(s). The choice of η 0 corresponds to the initial particle labelling and represents a gauge freedom in the problem. Equation (1.10) automatically incorporates the dynamic boundary condition (1.8b) when we pull-back the problem onΩ Since the flow is spherically symmetric, η is parallel to the vectorfield y. We introduce the ansatz η(s, y) = χ(s, r)y, r = |y|, r ∈ [0, 1], (1.12) and denote χ(0, r) by χ 0 (r). The Jacobian determinant of Dη expressed in terms of χ takes the form J [χ] := χ 2 (χ + r∂ r χ). (1.13) Since ∂ s J = J (divũ) • η, as a consequence of the continuity equation the Eulerian densityρ evaluated along the particle world-lines satisfies The fluid enthalpy is a function r → w(r) defined through the relationship w(r) α =ρ 0 (χ 0 (r)r)J [χ 0 ](r), (1.16) and it is a fundamental object in our work. Instead of specifyingρ 0 and χ 0 , throughout the paper we fix the choice of the fluid enthalpy w satisfying properties (w1)-(w3) below.
(1. 19) The purpose of the following lemma is to show that there exist choices of the enthalpy w consistent with the above assumptions. Lemma 1.1. For any n ∈ N there exists a choice of the enthalpy w satisfying properties (w1)-(w3). In particular, the resulting map g defined by (1.18) satisfies (1.19).
Remark 1.2. It is evident from the proof that one can easily modify the enthalpy w in the regions away from r = 0 so that (1.19) is still satisfied. In fact, the family of enthalpies w which satisfy the assumptions (w1)-(w3) is infinite-dimensional.
As a simple, but important corollary of (w3), specifically (1.19), we have Corollary 1.3. Let g be given by (1.18). Then the following properties hold: (i) the map r → g(r) is monotonically decreasing on [0, 1]; (ii) in the vicinity of the origin the following Taylor expansion for g holds: g(r) = g(0) − c n r n + o r→0 (r n ) (1. 20) for some constant c > 0; (iii) for any k ∈ N there exists a positive constant c k such that (r∂ r ) k g(r) ≤ c k r n . (1.21) As shown in [33], the momentum equation (1.7b) expressed in the Lagrangian variables (s, r) reduces to a nonlinear second order degenerate hyperbolic equation for χ: where r → G(r) is given above in (1.17) and the nonlinear pressure operator P is given by (1. 23) We may explicitly relate the Eulerian density, the fluid enthalpy and the Jacobian determinant; as long as J [χ] > 0 by (1.14) and (1.16) we have the fundamental formulã ρ(s, χ(s, r)y) = w α (r)J [χ] −1 .
(1.24) Remark 1. 4. Without being precise about the definition of the gravitational collapse for the moment, our goal is to prove that there exists a choice of initial conditions with a particular choice of the enthalpy w so that J [χ] becomes zero in finite time. We shall then show that there indeed exists a densityρ 0 satisfying the physical vacuum condition as both the profile w α and the initial labelling of the particles χ 0 are necessary to recover the Eulerian densityρ 0 , see (1.16).
Remark 1.5. In the special case when χ 0 = 1, w α andρ 0 coincide. We refrain from imposing the initial condition χ 0 = 1, but we shall prove a posteriori that the initial conditions that we use for the construction of the collapsing stars indeed satisfy χ 0 = 1 + O(ε) in a suitable norm.

Pressureless collapse
The first step in our analysis is to describe the solutions of (1.22) when ε = 0. We are led to the ordinary differential equation (ODE) with initial conditions χ(0, r) = χ 0 > 0, χ s (0, r) = χ 1 . (1.29) We now give a detailed description of the dust collapse from both the Lagrangian and Eulerian perspective, as this will serve as the leading order description of the collapsing stars for the EP γ system.
Notice that for any fixed r ∈ [0, 1] the coefficient G(r) merely serves as a parameter in the above ODE. The total energy is clearly a conserved quantity. We are interested in the collapsing solutions, i.e. solutions of (1.28)- (1.29) such that there exists a 0 < T < ∞ so that lim s→T − χ(s, r) = 0 for some r ∈ [0, 1]. We consider the inward moving initial velocities with χ 1 < 0. From the conservation of (1.30) we obtain the formula (1.31) Integrating (1.31) one sees that for every r there exists a 0 < t * (r) < ∞ such that χ(t * (r), r) = 0. A simple calculation reveals that for any r ∈ [0, 1] we have the universal blow-up exponent 2/3 χ(s, r) ∼ c(r)(t * (r) − s) Observe that the Eulerian description of the solution seizes to make sense at and after time s ≥ t * .
On the other hand, for different values of r the Lagrangian solution may make sense even after t * .
In particular, when t * (r) is a non-constant function, we can speak of a "fragmented" or "continued" collapse, wherein particles with a different Lagrangian label r collapse at different times. This is the hallmark behavior of inhomogeneous collapse. For simplicity, we shall consider a special subclass of solutions of (1.28)-(1.29) with zero energy. Up to multiplication by a constant such profiles have the form χ dust (s, r) = (1 − g(r)s) 2 3 , (1. 33) where g is given by (1.18). It follows that χ dust becomes zero along the space-time curve srg ′ (r) 1 − g(r)s , (s, r) ∈ Ξ.
Since r → g(r) is monotonically decreasing, particles that start out closer to the boundary of the star take longer to vanish into the singularity.

Main theorem and related works
Stellar collapse is one of the most important phenomena of both Newtonian and relativistic astrophysics. Even though extensively studied in the physics literature, very little is rigorously known about the compactly supported solutions to EP γ -system that lead to the gravitational collapse.
(1) When P (ρ) = 0 and therefore the star content is the pressureless dust, there exists an infinitedimensional family of collapsing dust solutions, as described in Section 1.2.
(2) If γ = 4 3 in (1.2), due to the special symmetries of the problem, "homologous" self-similar collapsing solutions exist and were discovered by Goldreich and Weber [24] in 1980. Further rigorous mathematical works about such solutions are given in [42,23,21]. Here all the gas contracts to a point at the same time and the dynamics is described by a reduction to a finitedimensional system of ODEs.
(3) When γ > 4 3 it is shown in [20] that the collapse by density concentration cannot occur. We refer to the values 1 < γ < 4 3 , γ = 4 3 , and γ > 4 3 of the adiabatic exponent as the mass supercritical, mass-critical, and mass subcritical cases respectively. This terminology is motivated by the invariant scaling analysis of the EP γ -system, see e.g. [28].
It has been an outstanding open problem to prove or disprove the existence of collapsing solutions in the supercritical range 1 < γ < 4 3 . Theorem 1.6 identifies an infinite-dimensional family of monotonically decreasing initial densities that lead to the gravitational collapse. This is a global characterization of the dynamics, as the region Ξ corresponds to the maximal forward development of the data at s = 0.
The best known class of global solutions to the EP γ system are the famous static Lane-Emden stars [10,1,61]. In the range 6 5 < γ < 2 one finds compactly supported radially symmetric timeindependent solutions of finite mass, whose stability still remains an outstanding open problem. In the subcritical range γ > 4 3 the question of nonlinear stability is open despite the promising conditional nonlinear stability result proven by Rein [47] (see [40] for rotating stars). If the solution exists globally in time when γ > 4 3 and the energy is strictly positive, then the support of the star must grow at least linearly in t, as shown in [43]. A similar conditional result holds when γ = 4 3 [20]. In the supercritical range 6 5 ≤ γ < 4 3 it has been shown by Jang [32,33] that the Lane-Emden stars are dynamically nonlinearly unstable. Besides the stationary states and the homologous collapsing stars in the mass-critical case γ = 4 3 , the only other global solutions of EP γ were constructed by Hadžić and Jang [28,30].
Since the works of Sideris [51,52] it has been well-known that solutions of the compressible Euler equation (without gravity) develop singularities even with small and smooth initial perturbations of the steady state (ρ, u) = (1, 0). This type of blow up is generally attributed to the loss of regularity in the fluid unknowns which typically results in a shock. Under the assumption of irrotationality, Christodoulou [14] gave a very precise information on the dynamic process of shock formation for the relativistic Euler equation. In the context of nonrelativistic fluids, a related result was given by Christodoulou and Miao in [15], while a wider range of quasilinear wave equations is treated extensively by Speck [55], Holzegel, Luk, Speck, and Wong [31]. Most recently shock formation results have been obtained even in the presence of vorticity by Luk and Speck [39], for an overview we refer the reader to [56]. A very different type of singular behavior which results in a wild nonuniqueness for the weak solutions of compressible Euler flows was obtained by Chiodaroli, DeLellis, and Kreml [12], inspired by the methods of convex integration, see [19] for an overview.
The above mentioned mechanisms of singularity formation are different from the singularity exhibited in Theorem 1.6, where the density and the velocity remain smooth in the vicinity of the origin and no shocks are formed before the gravitational collapse occurs.
In the absence of gravity, a finite dimensional class of special affine expanding solutions to the vacuum free boundary compressible Euler flows was constructed by Sideris [54,53]. Their support takes on the shape of an expanding ellipsoid. Related finite-dimensional reductions of compressible flows with the affine ansatz on the Lagrangian flow map go back to the works of Ovsiannikov [45] and Dyson [22], with different variants of the equation of state. Nonlinear stability of the Sideris motions was shown by Hadžić and Jang [29] for the range of adiabatic exponents 1 < γ ≤ 5 3 and it was later extended to the range γ > 5 3 by Shkoller and Sideris [49]. In the setting of compressible non-isentropic gaseous stars (where the equation of state (1.2) is replaced by the requirement p = P (ρ, T ), T being the internal temperature) it is possible to impose an affine ansatz (separation of variables) for the Lagrangian flow map and thus reduce the infinitedimensional PDE dynamics to a finite-dimensional system of ODEs. The resulting solutions have space-homogeneous gas densities and the system is therefore closed -the star takes on the shape of a moving ellipsoid. For an overview we refer to [3,4]. A number of finite-dimensional reductions in the absence of vacuum regions relying on self-similarity and scaling arguments can be found in the physics literature e.g. [38,46,50,60,58,5,2,6,48].
Without the free boundary, in the context of finite-time break up of C 1 -solutions for the gravitational Euler-Poisson system with a fixed background we refer to [11] and references therein. There are various models in the literature where the stabilizing effects of the pressure are contrasted to the attractive effects of a nonlocal interaction; we refer the reader to [9,7,8] for a review and many references for different choices of repulsive/attractive potentials.
The analogues of the collapsing dust solutions in the general relativistic context were discovered in 1934 by Tolman [59]. In their seminal work from 1939, Oppenheimer and Snyder [44] studied in detail the causal structure of a subclass of asymptotically flat Tolman solutions with space-homogeneous density distributions, thus providing basic intuition for the concept of gravitational collapse. Nevertheless, in 1984 Christodoulou [13] showed that the causal structure of solutions described in [44] is in a certain sense non-generic in the wider family of Tolman collapsing solutions, proving thereby that for densities given as small inhomogeneous perturbations of the Oppenheimer-Snyder density, one generically obtains naked singularities. This in particular highlights the importance of the rigorous study of the gravitational collapse of gaseous stars with more realistic equations of state, i.e. with nontrivial pressure. In the absence of any matter, existence of singular solutions containing black holes has been known since 1915. This is the 1-parameter family of Schwarzschild solutions, which is embedded in the larger family of Kerr solutions. The nonlinear stability of the Kerr solution has been an important open problem in the field. Substantial progress has been made over the recent years by Dafermos, Rodnianski, Holzegel, Shlapentokh-Rothman, Taylor, see [17,18,57] and references therein.

Foliation by the level sets of χ dust
We would like to build a solution of (1.22) "around" the fundamental collapsing profile (1.33). To that end it is natural to consider the change of variables and introduce the unknown φ(τ, r) := χ(s, r).
The operator r∂ r expressed in the new variables is denoted by Λ and it reads We also use the abbreviation M g (τ, r) := (τ − 1)r∂ r (log g), From (1.22) we immediately see that the unknown φ solves is the pressure term in new variables (τ, r). In (τ, r)-coordinates the dust collapse solution (1.33) is denoted by φ 0 , it solves 48) and is given explicitly by After a simple calculation we obtain (1.50) In particular J [φ 0 ](τ, r) > 0 for all (τ, r) ∈ (0, 1] × [0, 1] and The connection between the above formulas and mass conservation for the dust solution is detailed in Section 1.2. From the formula (1.50), (1.44), and (1.19) we conclude that for 0 < τ ≪ 1 wherefrom the scale r n /τ emerges naturally and will play an important role in our work. We will prove Theorem 1.6 in the (τ, r)-coordinate system, using (1.46) as a starting point. This is natural, as the collapse surface in the new coordinates takes on a simpler description Γ = {τ = 0}.

Methodology and outline of the proofs
The continuity equation in Lagrangian coordinates reduces to (1.24), which implies that the blow-up points of the density coincide with the zero set of the Jacobian determinant Therefore, the key goal of this work is to identify a class of initial data that in a suitable sense mimic the bahavior of the dust solution and we do that by showing A natural idea is to consider the dynamic splitting where the relative remainder R is expected to be small in an appropriate sense. A straightforward calculation gives a partial differential equation satisfied by R, which at the leading order takes the schematic form, where one can show thatḡ The simplest way of interpreting the relative "strength" of each of the terms in (1.53) is to compute the associated energies by taking the inner product with ∂ τ R. A key term emerges The severe τ −3 -singularity has a bad sign and it cannot be controlled by the positive definite part of the natural energy. This issue is not a mere technicality, it is deeply connected with the focusing nature of the dust collapse and it is unlikely that the ansatz (1.52) can be successful. We note that we have already implicitly used the assumption γ < 4 3 via the scaling transformation (1.6), which resulted in the occurrence of the small parameter ε in (1.46). We want to further use γ < 4 3 , but with a more refined dynamic splitting ansatz. Namely, our main idea is to seek a more special solution φ of the form where φ app will be chosen as a more accurate approximate solution of the Euler-Poisson system (1.46) in hope of mitigating the issue explained above. The exponent m > 0 is a sufficiently large positive number, so that H is a weighted remainder, small relative to τ m = φ Step 1. Hierarchy and the construction of the approximate solution φ app (Section 2). We shall find the approximate profile φ app as a finite order expansion into the powers of ε around the background dust profile φ 0 , i.e. (1.55) With the solution ansatz (1.55) we can formally Taylor expand the pressure term εP [φ 0 + εφ 1 + . . . ] into the powers of ε, thus giving us a hierarchy of ODEs satisfied by the φ j : (1.56) Functions f j+1 , j = 0, 1, . . . , M are explicit and generally depend nonlinearly on φ k , 0 ≤ k ≤ j, and their spatial derivatives (up to the second order). The system of ODEs (1.56) can be solved iteratively as the right-hand side f j+1 is always known as a function of the first j iterates. To show that finite sums of the form (1.55) are good approximate solutions of (1.46), we must prove that the iterates φ j , j ≥ 1, are effectively "small" with respect to φ 0 . The mechanism by which this is indeed true is one of the key ingredients of the paper, in both the conceptual and the technical sense. In particular we shall have to choose special solutions of (1.56), as they are in general not unique (the two general solutions of the homogeneous problem are τ 4/3 and τ −1/3 ), which will allow us to see the above mentioned gain.
We now proceed to explain these ideas in more detail. To provide a quantitative statement, we assume that the enthalpy profile w satisfies in a neighbourhood of the center of symmetry r = 0. The exponent n ∈ N is our effective measure of flatness of the star close to the center. For a given γ ∈ (1, 4 3 ) we consider densities (1.57) with n so large that With this assumption in place we prove that the iterates {φ j } j∈N "gain" smallness and this conclusion is summarized in the following theorem: Theorem 1.8. Let M, K ∈ Z >0 be given. There exists a sequence {φ j } j∈{0,...,M} of solutions to (1.56) with φ 0 (τ, r) = τ 2 3 , constants C jkm depending on K and M , and a λ > 2 n such that for j ∈ {1, . . . , M } and ℓ, m ∈ {0, 1, . . . , K} we have (1.59) Therefore the iterates φ j exhibit a crucial gain of τ jδ with respect to the dust profile φ 0 = τ 2 3 ! This is one manifestation of the supercriticality (i.e. 1 < γ < 4 3 ) of the problem and it can be viewed as the gain of smallness in the singular regime 0 < τ ≪ 1.
To motivate (1.59), we explain informally how the gain happens for φ 1 . To find φ 1 we solve the ODE (1 + r n τ ) γ , r ≪ 1. We expect φ 1 to "gain" 2 powers of τ with respect to the right-hand side of (1.60), and thus with δ defined in (1.58). Of course δ can be positive if and only if γ < 4 3 and the exponent n from (1.57) is sufficiently large! Most important consequence of Theorem 1.8 is that it leads to a source term S (φ app ) generated by φ app (see Lemma 5.14) which satisfies a natural improved bound Therefore, if M ≫ 1, it is reasonable to expect that the remainder ansatz τ m H r (with m ≥ M δ) is consistent with our strategy of treating H as an error term, in the regime τ ≪ 1.
Another crucial input in (1.59) is the factor The gain is visible only in the asymptotic regime r n /τ ≪ 1, which suggests that the scale r n /τ plays a critical role in our problem. Indeed, this gain is important in the closure of the energy estimate for H -it is used to absorb various negative powers of r which inevitably appear in our high order energy scheme intimately tied to the assumed spherical symmetry.
The proof of Theorem 1.8 is complex and delicate. It is based on the introduction of special solution operators S 1 and S 2 (2.93)-(2.94) for the ODE (1.56). In addition to a careful and precise tracking of the powers of τ and r n τ , to see the gain of τ jδ one has to use different solution operators S 1 and S 2 for j ≤ ⌊ 2 3δ ⌋ and for j > [ 2 3δ ], respectively. The precise estimate (1.59) and the emergence of r n τ as a critical quantity is intimately tied to the algebraic structure of f j , j = 1, . . . , M , which in turn possesses a rich geometric information related to the Taylor expansion of the negative powers of the Jacobian determinant J [φ].

Step 2. Equation for the remainder H (Section 3).
Thanks to the crucial gain of τ jδ and in the presence of τ m factor, now H satisfies the following quasilinear wave-like equation: where at the leading order The precise formulas for the right-hand side F , g 00 , g 01 , and c[φ] are given in (3.162)-(3.164), (3.155) respectively. In comparison to (1.53), the remarkable new feature of (1.61) is the presence of the coefficient m(m−1) for m sufficiently large. This leads to a coercive positive definite control of the solution at the singular surface {τ = 0}.
Step 3. The physical vacuum and weighted energy spaces (Section 4) Much of the difficulty in producing energy estimates for (1.61) comes from an antagonism between two different singularities present in the equation.
• at τ = 0 the coefficient c[φ app ] and various others formally blow up to infinity. This is the singularity associated with the collapse at the singular surface τ = 0 and already explained above; • at r = 1 we have w = 0 and therefore the elliptic part of the quasilinear operator on the left-hand side of (1.61) does not scale like the Laplacian as r → 1. This is a well-known degeneracy associated with the presence of the vacuum boundary.
The assumption of physical vacuum can be recast as the requirement that the enthalpy w ≥ 0 behaves like a distance function when r ∼ 1, i.e.
Requirement (1.62) is important in establishing the well-posedness of (1.61). The local well-posedness theory for the physical vacuum problem was first developed in the Euler case [36,16], while the wellposedness statements for the gravitational Euler-Poisson system can be found in [33,41,25,28,30]. Nevertheless, the well-posedness theory cannot be directly applied to our setting, as (1.61) differs from the above mentioned works in two important aspects: the problem has explicit singularities at τ = 0 and the space time domain (τ, r) ∈ (0, 1] × [0, 1] is strictly larger than the domain (s, r) ∈ [0, 1 g(0) ) × [0, 1] which only covers the star dynamics up to the first stipulated collapse time t * = 1 g(0) , see Section 1.4.

Step 4. Energy estimates and the conclusion (Sections 5 and 6).
Since φ app ∼ φ 0 = τ 2/3 , τ -derivatives of φ app create severe singularities in τ as τ → 0, which leads to difficulties in our energy estimates. We must in particular abandon the use vector field ∂ τ to form the natural high-order energy and instead rely on purely spatial derivatives. Due to very precise and delicate features of the approximate solution φ app near the center r = 0 (as described in Theorem 1.8), we are forced to use polar coordinates throughout [0, 1], which results in the introduction of many novel analytic tools to control the singularity at r = 0.
To motivate the definition of high-order energy spaces we isolate the leading order spatial derivatives contribution from the left-hand side of (1.61): where D r := 1 r 2 ∂ r r 2 · is the radial expression for the three-dimensional divergence operator. This particular form of L α suggests that we have to carefully apply high-order derivatives to (1.61) in order to avoid singularities at r = 0. We therefore introduce a class of operators defined as concatenations of ∂ r and D r : and set D 0 = 1. The operators D j are then commuted with the equation (1.61). For some N sufficiently large, the idea is to form the energy spaces by evaluating the inner product of the commuted equation with D j H τ , j = 1, . . . , N . However, following the ideas developed in [36,28,29], we need to perform our energy estimates in a cascade of weighted Sobolev-like spaces. For any given j ∈ {1, . . . , N } the correct choice is the inner product associated with the weights w α+j . Definition 1.9 (Weighted spaces). For any i ∈ Z ≥0 we define weighted spaces L 2 α+i as a completion of the space C ∞ c (0, 1) with respect to the norm · α+i generated by the inner product and denote the associated norm by · α+i . Definition 1.10 (Weighted space-time norm). For any 0 < κ ≤ 1, N ∈ Z >0 , κ ≤ τ ≤ 1 we define the weighted space-time norm We see that the powers of the w-weights increase with the number of derivatives. Such spaces are carefully designed to control the motion of the free boundary at r = 1 and the key technical tool in our estimates is the Hardy inequality. This is natural since w ∼ 1 − r near r = 1. Similarly, the presence of τ -weights allows us to precisely capture the degeneration of our wave operator at the singular space-time curve {τ = 0}.
The positive function x → q ν (x) serves as a weight for the top order spatial derivative contributions in the above definition, with powers ν = − γ+1 2 and ν = − γ+2 2 respectively. Such weights appear in the dust Jacobian J [φ 0 ] and by means of expanding the true solution around φ 0 , functions q ν appear naturally in our energies. The presence of q ν highlights again the importance of the characteristic scale r n /τ in our problem. We prove the key theorem: Theorem 1.11 (The κ-problem). Let γ ∈ (1, 4 3 ) and m ≥ 5 2 be given. Set N = N (γ) = ⌊ 1 γ−1 ⌋ + 6. For a sufficiently large n = n(γ) ∈ Z >0 , there exist σ * , ε * > 0, M = M (m, γ, n) ≫ 1 and C 0 > 0, such that for any 0 < σ < σ * and any 0 < ε < ε * the following is true: for any κ ∈ (0, 1) and any initial data Theorem 1.11 gives uniform-in-κ bounds for the sequence H κ with initial data specified at time τ = κ. One may for example choose trivial data at τ = κ, i.e. set σ = 0 in the above theorem to generate a family of solutions {H κ } κ∈(0,1] . As κ → 0 we conclude the existence of a solution H on (0, 1]. By (1.54), this gives a solution φ = φ app + τ m H r of the original problem (1.46), thus allowing us to prove Theorem 1.6 (after going back to the (s, r) coordinate system). The proof of Theorem 1.11 is given in Section 5.6, while the proof of Theorem 1.6 is given in Section 6.  Many of our energy estimates depend crucially on both the gain of a τ δ -power and a power of r n τ in Theorem 1.8. The former allows us to obtain a crucial gain of integrability-in-τ close to the singular surface τ = 0, while the latter is needed to absorb negative powers of r arising from the application of the operators D j on φ app . This delicate interplay works out, but requires a certain "numerological" constraint, namely the coefficient n has to be large enough relative to the total number of derivatives N used in our energy scheme.
Despite the delicate tools and analysis, one term stands out and seemingly causes a major obstruction to our method. After commuting the equation with high-order operators D j and evaluating the (·, ·) α+j -inner product with D j H τ , an error term M [H] defined in (4.177) emerges. A simple counting argument suggests that the number of powers of w in M [H] is insufficient to close the estimates near the vacuum boundary, but we carefully exploit a remarkable algebraic structure within the term and obtain the necessary cancellation, see Lemma 5.8.
The last claim of Theorem 1.6 shows that the infinitesimal volume of the shrinking domain of our collapsing solution behaves like the infinitesimal volume of the collapsing dust profile. More importantly, using (1.24), one can conclude that the qualitative behavior on approach to the singular surface τ = 0 of the Eulerian densityρ is the same as that of the dust density, see Section 6.

Plan of the paper
Section 2 is devoted to the derivation of the hierarchy of ODEs (1.56) and the proof of Theorem 1.8. In Section 3 we derive the equation for the remainder term H. In Section 4 we introduce the high-order differentiated version of the H-equation derived in Section 3. We also define high-order energies that arise naturally from integration-by-parts and show (Section 4.2) that they are equivalent to norm S κ N from Definition 1.10. The remainder of the section is devoted to various a priori estimates and preparatory bounds. In Section 5 we prove the key energy estimates, culminating in the proof of Theorem 1.11 in Section 5.6. Finally, Theorem 1.6 is shown in Section 6. In Appendices A-C many important properties and analytic tools used in our estimates are shown. We present details of the product and chain rule within vector field classes P andP (Appendix A), commutator identities (Appendix B), and the Hardy-Sobolev embeddings (Appendix C). Finally, for the sake of completeness, we sketch the local well-posedness argument in Appendix D.

Notation
• By Z ≥0 , Z >0 we denote the sets of non-negative and strictly positive integers respectively.
• Writing A B means that there exists a universal constant C > 0 such that A ≤ CB. A B simply means B A. If we write A ≈ B we mean A B and A B.
• For a given a > 0 we denote the closed three-dimensional ball of radius a centered at 0 by B a (0).

The hierarchy
Formally we would like to build a solution of (1.46) as a sum of the approximate profile φ app (given as a finite series expansion in the powers of ε) and the remainder term θ which we hope to show to be suitably small. In other words, we are looking to write Plugging (2.66) into (1.46), we will now derive a formal hierarchy of ODEs satisfied by the functions φ j , j ∈ {1, . . . , M }. We define the source term S(φ app ): We first recall the formula of Faa Di Bruno (see e.g. [37]) which will be repeatedly used in this section. Given two functions f, g with formal power series expansions, where f n , g n and h n are constants with respect to x. Faa Di Bruno's formula gives where π(n, k) = {(λ 1 , . . . , λ n ) : An element of π(n, k) encodes the partitions of the first n numbers into λ i classes of cardinality i for i ∈ {1, . . . , k}. Observe that by necessity λ j = 0 for any n − k + 2 ≤ j ≤ n. Lemma 2.1 (Detailed structure of the source term S(φ app )). The source term S(φ app ) given by (2.67) satisfies are explicitly given by (2.82), (2.74) below, f 0 := 0, and withÕ j and h m given explicitly below by (2.75) and (2.80).
Proof. For any m ∈ N, ν ∈ R, there exists a smooth function R ε m,ν : R m → R such that where by the formula of Faa Di Bruno and R ε m,ν is smooth in a neighbourhood of 0, and for any ε ∈ (0, 1) R ε m,ν C ℓ ≤ C ℓ , for some constant C ℓ > 0 which grows as ℓ gets larger.
Our next goal is to expand the function J [φ app ] −γ in the powers of ε. To that end we first observe that where we note thatJ 0 = 1 since J 0 = J [φ 0 ]. The remainder R J is given by the formula We have and Similarly From the definition of the source term (2.67) in therefore follows with f 0 := 0 and as claimed.
Motivated by the previous lemma, we define the hierarchy of ODEs where φ 3 0 = τ 2 , f j is given by (2.84) andÕ j and h j given by (2.75) and (2.80) respectively.
With {φ j } j=1,...,M satisfying (2.85), Lemma 2.1 in particular implies that given by (2.82), (2.74). Therefore, by solving the hierarchy up to order M we force the source term to be of order ε M+1 .

Solution operators and definition of
The number N will later correspond to the total number of derivatives used in our energy estimates.
Definition 2.2 (The "gain" δ and δ * ). Let γ ∈ (1, 4 3 ) be given and letγ = 4 3 − γ. For any natural number n > N +2 2γ we define 3 ) be given and fix an arbitrary natural number a ∈ Z >0 . Then there exists an n * = n * (γ, a) such that Proof. For the simplicity of notation let j := 2 3δ(n) . Then it is easy to check that (2.90) is equivalent to . (2.91) Since 1 < 1 3γ it is clear that the above inequality will be true if n is chosen sufficiently large.

Remark 2.4. Lemma 2.3 implies in particular
A simple calculation shows that the previous ODE is formally equivalent to This motivates the following definition of the solution operators: (2.95) The above definition of the solution is designed to enforce the gain of τ δ with respect to the previous iterate for all j ∈ {1, . . . , M }. Since M ≫ ⌊ 1 3γ ⌋, the above choice of the formula at the index values j > ⌊ 1 3γ ⌋ is crucial, see Proposition 2.8 and Lemma 2.14.
2.2 Bounds on φ j and proof of Theorem 1.8 The main goal of this section is the proof of Theorem 1.8. To that end, we need a number of preparatory steps. We first introduce the notation For the remainder of the section, constants M, K ∈ Z >0 are arbitrarily large fixed constants.
Lemma 2.7 (Basis of the induction). Let φ 1 be given by (2.95). Then The main result of this section is the quantitative estimate on the space-time derivatives of the iterates φ j . Proposition 2.8 (Inductive step). Let φ j be given by (2.95). Let 1 ≤ I < M be given and assume that for any j ∈ {1, . . . , I} and any ℓ, m ∈ {0, 1, . . . , K} we have Then for any ℓ, m ∈ {0, 1, . . . , K} the following bound holds where λ = 2N n is given in Definition 2.5. Remark 2.9. The constants in the above statement depend on K, M ∈ Z >0 and they generally grow as K and M get larger.
Proofs of Lemma 2.7, Proposition 2.8, and finally Theorem 1.8 are contained in Section 2.2.2. Before that we need a number of auxiliary bounds.
Proof of (2.103) The bound is obvious.
Proof of (2.104) We use the formula of Faa Di Bruno. We may write

Formula of Faa Di Bruno then gives
where we refer to (2.70) for the definition of π(m, j). Since Proof of (2.105) Using the formula of Faa Di Bruno like above, replacing formally ∂ m τ by (r∂ r ) ℓ we obtain where we refer to (2.70) for the definition of π(ℓ, k). Therefore, for ℓ ∈ Z ≥1 , by using the Leibniz rule where we have made use of (2.102). From this bound, the product rule, and (2.118), we conclude where we have used (2.102), (2.119), (2.96), the identity ℓ i=1 λ i = j which follows from the definition of the index set π(ℓ, j), and the trivial estimate x ≤ q 1 (x). Proof of (2.106) By letting ϕ = φ j , j ∈ {1, . . . , I}, in (2.109) we obtain where we have used the inductive assumption (2.99). If j = 0, from (2.110) we have Recalling J k from (2.77), applying the Leibniz rule and using (2.99) and (2.121)-(2.122) we obtain Proof of (2.107). We use the Faa Di Bruno formula again. Analogously to (2.118) we obtain where we have made use of (2.106). Using this bound just like in (2.120), we obtain (2.107).
Proof of (2.108). Using the formula of Faa Di Bruno, for any k ∈ {1, . . . , I} we have where we have used the inductive assumption (2.99), identities , and the additive property of p µ,ν .
1. Let 0 < λ < 1 be given and let β satisfy Then the following bound holds Then the following bound holds Proof. Proof of part (i). Applying the change of variables x = τ ′ /r n we havê where the very last inequality follows from 1 p λ,0 ( r n τ ) which in turn relies on r n ≥ τ .
where the very last inequality follows from 1 1 (1+ r n τ ) λ which in turn relies on r n ≤ τ .
Proof of part (ii). By the same change of variables as in (2.130) we havê (2.131) We distinguish two cases again. Case 1: r n ≥ τ . We have from b + 2 n < −1, where the last inequality follows from 1 p λ,0 ( r n τ ) which in turn relies on r n ≥ τ , just like in Case 1 in part (i). Case 2: r n ≤ τ . We have where the very last inequality follows from 1 ≤ 1 (1+ r n τ ) λ which in turn relies on r n ≤ τ . The two previous estimates together with (2.131) give (2.129).

Proofs of Proposition 2.8, Lemma 2.7, and Theorem 1.8
Proof of Proposition 2.8. We first assume that m = 0. Let k ∈ N and |f | τ 4/3 , |g| where we have first used (2.128) and then (2.129). Note that we have used the assumption k ≤ 1 3γ and Lemma 2.3 to ensure that − 5 3 + kδ + 2 n < −1 and therefore (2.129) is applicable in the last line of (2.132). If k > 1 3γ we then have When m = 1 we observe by taking τ derivative of (2.93) and (2.94) and by Lemma 2.12, 2.14 Similarly, using the Leibniz rule like above, For m ≥ 2 we simply use the equation where we have used the inductive assumption (that has been verified for all m ′ < m) and

Remainder equations and the main results
We look for a solution of (1.46) in the form where M is to be specified later.

Derivation of the remainder equations
Lemma 3.1 (PDE satisfied by θ). Let φ, φ app , and θ be related by (3.137). Then the equation satisfied by θ reads where the source term S(φ app ) and the expressions K j [θ], j = 1, 2, 3 are given by (2.67), (3.151), (3.152), (3.153) below respectively and c is given by (3.155 We want to find alternative expression for where we have used the identity 3φ 2 app + 6φ app θ + 3θ 2 = 3φ 2 . We may rewrite Similarly, We may write where Plugging (3.146) and (3.147) into (3.141), we deduce that the last line of (3.149) can be rewritten as which asserts that the expression is a nonlinear term. Therefore by splitting we obtain and where the source term S(φ app ) is given by (2.67) and Then H solves where the source term S(φ app ) and the expressions K j [θ], j = 1, 2, 3 are given by (2.67), (3.151), (3.152), (3.153).
Proof. The proof follows by a direct verification after plugging in θ = τ m r −1 H in (3.154).
We rewrite (3.157) in the form The leading order operator will be shown to be hyperbolic due to the bound 1 g 00 1 shown later in Lemma 4.6. We shall see that the former estimate is crucially tied to the supercriticality (γ < 4 3 ) and the flatness assumption on the enthalpy w near r = 0 (i.e. n sufficiently large in (1.19), i.e. Lemma 1.1). Moreover, ✷ is also manifestly quasilinear as c[φ] depends on the space-time derivatives of H. The twofold singular nature of ✷ coming from the gravitational singularity at τ = 0 and the vacuum singularity at r = 1 is discussed at length in Section 1.5.
The basic equation for our energy estimates is obtained by dividing (3.162) by g 00 : We denote the first summation without sup in Definition 1.10 of S N κ by E N and the second summation without the time integral by D N , i.e. for any τ ∈ (0, 1] we let Then the space-time norm can be written as 4 High-order energies and preparatory bounds

High-order equations and energies
In order to derive high-order equations, we first introduce the elliptic operators Then for any f, h we have where we recall the inner product (·, ·) k given in (1.65). We recall here the definition of the fundamental high-order differential operators D j given in (1.64). We then define Important role is played by the operatorD i defined as Here C i contains all the commutators where the functions ζ ij are given by (B.414) and the commutators [·, ·] are defined in (B.415). Furthermore, (4.177) Note that we have written for i ≥ 1, Definition 4.1 (Weighted high-order energies). For any 0 < κ ≤ 1 and N ∈ N we define the high-order energies where for any 0 ≤ j ≤ N we have g 00 w |D j+1 H| 2 w α+j r 2 dr (4.179) and

A priori bounds and the energy-norm equivalence
Assume that H is a solution to (3.162) on a time interval [κ, T ] for some T ≤ 1. For a sufficiently small σ ′ < 1, to be fixed later, we stipulate the following a priori bounds.
Proof of (4.210). Observe the identity where we have used (1.26). Since 1 g, G 1, the claim follows.
A corollary of Lemma 4.6 is the proof of equivalence between the norms and energies given respectively by Definitions 1.10 and 4.1.

Vector field classes P andP
We now introduce a set of auxiliary, admissible vector fields associated with differential operators D i andD i that allow us to circumvent coordinate singularities near the origin and to obtain high order estimates effectively. They are obtained by allowing 1 r in addition to D r whenever D r appears in the chains of D i andD i . In other words, for j ≥ 0 and set P 0 = {1}. Likewise, we definē In what follows, we derive the bounds ofP of various quantities involving φ app , φ, φ + Λφ and so on that will be useful for the high-order energy estimates.
Lemma 4.8. The following bounds hold true: where we have used Proposition 2.8 in the second line.
Recall that Λφ app = M g ∂ τ φ app + r∂ r φ app . By Lemma A.7, definition (1.44) of M g , and the property (1.21) we obtain where we have used the same argument as in the proof of (4.214) to obtain the second summand in the last bound above.
A simple consequence of Lemma 4.8 is the following corollary.
Corollary 4.9. The following bounds hold true:
Proof of (4.227). Recalling (1.47) it is easy to check that (4.228) We now apply the product rule (A.403) and bounds (4.223), (4.222), (4.214) and the estimate |w ′ | r n−1 to conclude above, we obtain the claim, where in particular we use γ > 1.

Preparatory bounds
Proof. Bound (4.229) follows directly follows from Further For the first term, we use (A.399) to rewrite Therefore we deduce that It is easy to see that Putting together the above bounds we obtain (4.230).
The same conclusions hold in Lemma 4.11 when we replaceD i by any V ∈P i .
Applying the product and the chain rule, for any can be written as a linear combination of In order to estimate V j φ and W j (φ+Λφ), it suffices to estimateD i φ andD i (φ+Λφ).
We first consider k * = a ℓm ≥ i jm . Let j m0 + 1 be the first index for which i jm 0 +1 ≥ 2 so that i j = 1 for j ≤ j m0 . In this case, we rearrange the w-weight in (4.253) as follows: The goal is to estimate the last term w α+2a ℓm +2−N 2 W ℓm (φ+Λφ) φ+Λφ in L 2 -norm and all the remaining ones in L ∞ . Note that Therefore, the exponent of the first w in the second line is non-negative and therefore, that factor is bounded. Now all i j 's and a ℓ 's except a ℓm cannot be bigger than N − 3, otherwise, it would contradict the definition of k * . Thus we can apply (4.237) and (4.243) to the first line above. Moreover, since 2 ≤ a ℓm ≤ N − 1, we can apply the weighted L 2 -embedding (4.245) to the W ℓm term in the second line of the right-hand side of (4.255). By (4.248) q c ( r n τ )φ a+2b (φ+Λφ) b L ∞ ≤ τ 2 3 a+b since b+c ≤ 0 by our assumptions. This gives the bound The case k * = i jm > a ℓ can be treated in the same fashion where we use (4.239) instead of (4.245). Proof of (4.250). In this case, since k * ≤ N − 3 and as above we first consider the case k * = a ℓm . We then have Note that jm j=jm 0 +1 (i j − 2) + ℓm ℓ=1 (a ℓ − 1) ≤ i and therefore, the exponent of the first w in the nextto-last line above is non-negative as before. Using (4.236)-(4.237), (4.242)-(4.243), and the bound q c ( r n τ )φ a+2b (φ+Λφ) b L ∞ ≤ τ 2 3 a+b , we can bound all the remaining factors to finally obtain (4.250). Proof of (4.251). When i = N and k * ≤ N − 1 we may use the already proven (4.249) to infer that the · α+N +2 -norm of (4.253) is bounded by the right-hand side of (4.251). It now remains to discuss the case k * = N in which case either 1) j m = 1 and i jm = N (a ℓm = 0) or 2) ℓ m = 1 and a ℓm = N (i jm = 0). When a ℓm = N , the expression (4.253) reads and therefore by (4.246) and (4.248), we deduce as claimed. When i jm = N , the corresponding estimate reads where we have used (4.240).
We conclude the section with several a priori estimates that will be important for the energy estimates in Section 5.
Lemma 4.15. Recall g 00 defined in (3.163). The following bounds hold: Proof. It suffices to prove the bounds for V g 00 as the corresponding bound for V ( 1 g 00 ) is a simple consequence of the chain rule (A.405) and the bound (4.204). From (3.163) and (3.155) it follows that for any V ∈P i with i ≥ 1 we have In particular, if ℓ 2 ≤ i − 1 we may estimate Using Lemma 4.14 now with c = 2 − 2 n (< γ + 1), estimates (4.259)-(4.261) follow easily. If ℓ 2 = i and additionally i ≤ N − 1 we may still run the same argument. If however ℓ 2 = N , we lose √ ε in (4.262) due to (4.251).

Since by (3.164) for any
by an analogous argument we have the following lemma: Lemma 4.16. Recall g 01 defined in (3.164). The following bounds hold:

Energy estimates
To facilitate our proof and carry out the energy estimates, for the remainder of this section we assume that H be a solution to (3.162) on a time interval [κ, T ] for some T ≤ 1, the a priori assumptions (4.182) hold, and the following (rough) bootstrap condition is true:

Estimates for L low -terms
The goal of this section is the following proposition.

Decomposition of L low H
We rewrite the linear operator L low in the form We now use (4.227) and the L 2 -embeddings (C.427) if ℓ 2 ≥ 3 and otherwise (C.429) to conclude where we have used the bound w α+i w α+2i−N w α+2ℓ2−N . We now focus on the second term in the first line of (5.271).
Case I-2. ℓ 3 < ℓ 4 . If ℓ 4 ≤ 2 then we are in the regime that has already been discussed above. Assume ℓ 4 ≥ 3. If ℓ 3 ≥ 2 we use (4.250) and (C.427) to obtain We have used the inequality i − 2(ℓ 3 + ℓ 4 ) + N ≥ N − i ≥ 0. The case ℓ 3 ≤ 2 is handled similarly, with (4.248) instead of (4.250). Case II. ℓ 3 = i. In this case we need to bound L 2 respectively. The remaining terms are estimated in L ∞ and we conclude that the expression in (5.282) is bounded by ετ δ * (D N ) 1 2 just like above. If however i = N we must use (4.251). It then follows that the expression in (5.282) is bounded by The 3rd-7th term in (5.271) are estimated analogously. Note that the terms ∂ τ H and H τ and similarly ∂τ H τ and H τ 2 are on equal footing from the energy stand point or more precisely where we recall the definitions (3.168)-(3.169) of E N and D N . In particular, the estimates for the 3rd, 5th, and the 7th term in (5.271) are very similar and we sketch the details for the 7th (next-to-last) term. By the product rule (A.403) we have A case-by-case analysis analogous to the one above, Lemma 4.14, and Lemmas C.3-C.5 yield The same bound, with Hτ τ replaced by H τ 2 follows analogously. The 4-th and the 6-th term on the right-hand side of (5.271) are easier to bound. In the 6-th term the factor w ′ gives a regularising power of r near the center r = 0 due to the bound |∂ a r w ′ | ≤ r n−a−1 (which in turn follows from (1.19)). Similarly, the presence of ΛJ [φ app ] in the 4-th term, by virtue of (4.222) affords a power of ( r n τ ) in our estimates, which again counteracts any potential singularities coming from the negative powers of r near r = 0. Routine application of Lemmas C.3-C.5 and Lemma 4.14 yields the desired bound.
To estimate the last line in (5.271) we first observe that We can therefore break up the last line of (5.271) into a sum of terms that are of similar structure as the ones showing up above, and thus the estimate follows analogously and thus obtain the same bound as in (5.283).
Proof. We focus on the first and the most complicated term on the right-hand side of (5.272). Recall that r∂ r H r = D r H − 3 H r . By analogy to (5.277) we have Case I-1. ℓ 3 = i. In this case ℓ 1 = ℓ 2 = ℓ 4 = 0 and we note that where we we have used (4.215). Therefore we bound the · α+i norm of the last line of (5.287) by If i = N we haveD N D r = D N +1 . Therefore, by (C.428), w α+N +2 w α+N +1 , and q − γ+1 On the other hand, using (A.401) we additionally have Plugging the last bounds into the last line of (5.291) and recalling (2.88) we bound it by √ ετ instead of (C.428) above and obtain the upper bound Recalling (4.215) and the bound |A 1 ( w g 2 rg 00 )| r −1−ℓ1 , by (5.288) we have If ℓ 4 ≤ 1 and therefore ℓ 3 ≤ 1,we can estimate the · α+i -norm of the last line of (5.293) using (4.248) and (C.430) by We rely on (C.427) and (4.250) to bound the · α+inorm of the last line of (5.293) by ∞ by a constant. If on the other hand ℓ 3 + ℓ 4 = i, then ℓ 1 = 0 and therefore we have an additional power of w in our estimate which by the same idea as above allows us to obtain the bound (5.295).
This case is handled analogously to the case II-1 above and relies on a similar case distinction (ℓ 4 ≥ 2 and ℓ 4 ≤ 1) as well as Lemma 4.14 and estimates (C.429), (C.427).
This completes the bound of the first term on the right-hand side of (5.272). The estimates for the remaining 2 terms proceed analogously. Note that we use (4.224) crucially to estimate the third term on the right-hand side of (5.272).

High order commutator estimates
The goal of this section is the following proposition.
Proof. Proof of (5.297). By (B.416) we have the formula ετ δ * −1 by Lemma 4.16, the bound w α+i w α+2i−N , and definitions (2.88)-(2.89) of δ and δ * , we have In order to bound the second term on the right-hand side of (5.300) we distinguish several cases by analogy to Lemma 5.2. Case I: ℓ 1 ≤ ℓ 2 . If ℓ 2 ≤ 2 and therefore ℓ 1 ≤ 2, we can use (C.429) and (4.264) to obtain If 3 ≤ ℓ 2 ≤ N we again distinguish 2 cases. If ℓ 1 ≥ 2 we can use (C.427), (4.266), and (2.88)-(2.89) to obtain If ℓ 1 = 1 we then use (4.264) instead of (4.266) and obtain the same conclusion. Case II: ℓ 1 ≥ ℓ 2 . In this case we proceed analogously and rely crucially on Lemmas 4.16 and estimates (C.429)-(C.431). The only nonstandard situation occurs when ℓ 1 = N . In that case ℓ 2 = 0 and we must use the bound (4.267) together with (C.429). We then obtain To estimate the last term on the right-hand side of (5.300) we note that for any A 2 ∈P ℓ2 , we have A 2 D r ∈ P ℓ2+1 and since ℓ 2 ≤ i − 2 we are in the regime treated above. This concludes the proof of the bound for D i , g 01 g 00 ∂ r ∂ τ H α+i . The remaining 2 terms on the left-hand side of (5.297) are estimated analogously and their proofs rely crucially on Lemmas 4.15 and 4.16. The second term is less singular with respect to τ and the presence of the g 01 does not change the structure of the estimates due to Lemma 4.16. The third term contains the factor H τ 2 which, from the point of view of the energy, scales just like Hτ τ and thus the structure of the estimates is similar to the above. Proof of (5.298). From (4.170) we have By the commutator formula (B.418) we have The second sum on the right-hand side of (5.306) can be estimated analogously to the estimates for (5.297) above, using (5.305). Thereby we observe that the total number of derivatives in the operator A 2 D r L α is at most i, since ℓ 2 ≤ i − 3. We next focus on the first term on the right-hand side of (5.306). SinceD i−2 D r = D i−1 , using (5.305) we can write it as By the product rule (A.403) we can isolate the top-order term We now use (4.248), (C.428) and conclude, in the case i = N where the estimate (4.248) has been used in the third line. When using (4.248), we first recall (3.155) and use the product rule to write We note that by Lemma 4.15 and (1.19) we have 1 g 2 g 00 + ∂ r 1 g 2 g 00 1 and therefore (4.248) yields the third line above. The remaining below-top-order terms can be estimated analogously to (5.297) to finally obtain (5.298).
Proof of (5.299). Since ∂ k r w r n−k for any k ∈ {1, . . . , n} it follows from (B.414) |ζ ij | r n−j−2 . Therefore by (4.248) for any j ≤ i we have Therefore for any j ≥ 3 we have where we have used (C.427) in the second line. If j ≤ 2 we use (C.429) instead of (C.427) and obtain the same bound.

High-order estimates for M [H]
We first recall K a [θ], a ∈ R in (3.139): and also Lemma 5.6. We have the following bound: Proof. First we recall that (3.139) implies which together with (C.430) and θ = τ m H r yields or equivalently, The representation (5.310) then gives (5.311) or equivalently . We start with a = 1. By applying the product rule to (5.314) and using (C.430), (C.431) and (C.433), (4.214), (4.215), we deduce that For general a ∈ R, let us write down the expression forD j K a [θ]. For j = 1, using (3.147), we haveD For j ≥ 2, by applying the product rule and chain rule, we deduce that which can be proved based on the induction argument on j. Therefore, we deduce (5.312). Also, by (C.427) we have (5.313).
Before we proceed with the estimates, we examine the structure of M [H]. Recall (4.177) and the formula (3.155 where we have used (4.170) and written L α H = −(1 + α)w ′ D r H − w∂ r D r H. Our source of concern is rectangular bracket above, as it contains top-order terms with two derivatives falling on H (either ) and seemingly insufficient number of w-powers to allow us to bound them through our w-weighted norms. This situation is a typical manifestation of the vacuum singularity at the outer boundary. Our key insight is that, due to special algebraic structure of the equation, the terms involving two spatial derivatives of H without the corresponding multiple of w will be cancelled. In the following lemma, we present the rearrangement of M [H] that elucidates such an important cancelation.

(i) M [H] can be rewritten into the following form
where K 3 is defined by (3.153). (ii) Each expression in the right-hand side of (5.322) contains at most two spatial derivatives. If two spatial derivatives of H appear in the expression, they always contain a factor of w. In particular, the last bracket of the first line of K 4 [θ] in (5.323) can be rewritten without any second spatial derivatives of H: Proof. To verity (5.322), we will first rewrite the rectangular bracket in (5.321). By (3.158 where we have used r∂ r H r + 3 H r = D r H and θ = τ m H r . For the first term in the rectangular bracket in (5.321), we note Together with (5.325), the rectangular bracket in (5.321) gives rise to the first line and the first term of the third line of (5.322). The following line of (5.321) corresponds to the second line of (5.322) where we have used L α H = −(1 + α)w ′ D r H − w∂ r D r H.
Finally we will count the number of spatial derivatives and the weight w. First of all, it is clear that all the terms appearing in (5.322) contain at most two spatial derivatives of H. For instance, the first term in (5.322) does not contain the second derivatives of H. In the second line, both terms contain the second derivatives of H and they have a factor of w. Note that ∂ r g 00 has a term involving two derivatives (see (3.163)) but that comes with w. The same counting applies to the rest. The only expression that is not obvious at first sight is the first line of (5.323) because we do see the two spatial derivatives of H without the weight w. It turns out that those second derivatives disappear after cancelation. A direct computation using (5.314) yields the identity (5.324).
Lemma 5.9. Let H be a solution to (3.162) on a time interval [κ, T ] for some T ≤ 1 and assume that the a priori assumptions (4.182) holds. Then Proof. We note that the terms in the first two lines of (5.322) have the similar structure as the terms resulting from D r ( 1 g 00 L 2 low H) in terms of the highest order derivative count and the weight w count. For instance, the first line of (5.322) is comparable to the case when the derivative falls into w of the last term of (5.272). The difference is whether the coefficients are set by φ app , J [φ app ] or φ, J [φ], but the coefficients enjoy similar bounds due to Lemmas 4.8 4.10 for φ app , Lemma 4.14 for φ and our a priori assumption (5.268). We therefore have where M 12 denotes the first two lines of (5.322). We focus on the last line of (5.322) and present the detail for the bound on We restrict our attention first to the following term coming from the first line of (5.323) As shown in the previous lemma, the identity (5.324) assures that ∂ r K 1 [θ] − φ 2 [r∂ 2 r θ + 4∂ r θ] contains at most one spatial derivative of H and therefore no issues associated with the w-weights near the boundary will occur.
We proceed withD i−1 (⋆) for 1 ≤ i ≤ N . By the product rule,D i−1 (⋆) can be written as a linear combination of the following form As before, we divide into several cases. If ℓ k ≤ 2 for all k = 1, 2, 3, all the indices are low and we just use L ∞ bounds (4.236), (4.259), (4.261), (5.312). In the following, we assume that at least one index is greater than 2. Case I: ≤ N − 4, and we apply L ∞ bounds for A 1 and A 2 factors and L 2 bounds for A 3 factor. In particular, by assuming ℓ 1 ≥ 1 (the case of ℓ 1 = 0 follows similarly), we arrange the w weights as follows: By the product rule and by using (4.236), (4.237), (4.259), (4.261), we deduce that and by further using (5.312) We have derived so far We claim that Note that from (5.324) we may rewrite r{∂ r K 1 [θ] − φ 2 [r∂ 2 Using (5.320) for a = −γ − 1, we have the following expression Following the case-by-case analysis as before and using (5.311), (5.312), (5.313), Lemma 4.14 and (4.223), we deduce that Therefore we obtain the same bound as Case I where we have used (4.260). Case III: ℓ 1 ≥ max{ℓ 2 , ℓ 3 }. In this case, 3 ≤ ℓ 1 ≤ i − 1 and ℓ 2 , ℓ 3 ≤ i−1 3 . We apply L ∞ bounds for A 2 and A 3 factors and L 2 bounds for A 1 factor. For L 2 bounds for A 1 ( 1 g 00 ), we use Lemma 4.15. The proof follows in the same fashion and we get the same bound as in the previous cases.
All the other terms in (5.323) are estimated analogously and we have the following bound: The last term in (5.322) can be estimated similarly by using Lemma 4.15, (5.325), and the previous estimates on K 4 [θ]: This finishes the proof Lemma.

Nonlinear estimates
Before we formulate the main estimate in Proposition 5.11, we collect several identities that can be regarded as a special form of the product rule that connects the algebraic structure of the nonlinearity to the algebraic properties of the vector field class P.
Note that each term in the summation belongs to P j for some j. Therefore, the claim is true for i = ℓ + 1. If ℓ is odd, we can rearrange terms as follows: which shows that the claim is true for i = ℓ + 1. The proofs of (5.341)-(5.342) are similar.
Proposition 5.11 (Estimates for the nonlinear term). Let H be a solution of (3.157). Then for any i ∈ {0, 1, . . . , N } the following bound holds Lemma 5.12. For each 1 ≤ i ≤ N − 1, we have the following: Proof. Using the product rule (A.403) we have where we recall the notation G = 1 r r∂ r H r 2 from the proof of Lemma 5.10.
Case II-2: ℓ ≥ 1, j = 0 and k ≥ 1. In this case, 2 ≤ ℓ * ≤ i and k ≤ N − 1 since ℓ + k = i ≤ N . If ℓ * ≥ k we have where we have used (4.250) and the embeddings (C.427) and (C.433). Moreover, w If ℓ * < k, as in Case I-2, we estimate w in the L 2 -norm and the appropriately weighted terms C 1 H and C 2 H in the L ∞ -norm and obtain the same bound as in (5.358).
Case II. ℓ 3 = i. If i ≤ N − 1 we proceed as in Case I. Assume now ℓ 3 = N . Since in this case A 3 =D N the last line of (5.362) takes the form We take special notice of the additional power of w available in this case. Since Mg g 2 r 2 τ 1− 2 n p 1,− 2 n ( r n τ )q 1 ( r n τ ) we can estimate the · α+N -norm of the above expression by where we have used (4.248), (C.428), the bound τ The third, fourth, and fifth lines on the right-hand side of (5.346) are easily bounded by the same ideas as above, where we systematically use the product rule (A.403), Lemmas 4.14 and 4.10.
We only highlight the potential difficulties and how they can be overcome. In the 3rd term on the right-hand side of (5.346) there is nothing dangerous; we may write it as εγτ m w φ 2 J [φ] γ+1 1 g 1 Λ(3φ app + Λφ app )H H r and then estimate its D i derivative using the case-by-case analysis analogous to the above, the product rule (A.403), Lemma 4.14, and the bounds (4.215), (4.220). Note that the last two estimates afford the presence of a power of r n τ in our bounds, which in turn has the regularising effect of diminishing any potential singularities due to negative powers of r at r = 0.
The 4-th term on the right-hand side of (5.346) looks potentially dangerous due to the presence of the negative powers of r in H 2 r 2 and H 3 r 3 . However, by (4.222) the bounds on V ΛJ [φ app ], V ∈P i , will afford a presence of a power of of r n τ , thus averting all difficulties with potential singularities at r = 0. Finally, the 5th term on the right-hand side of (5.346) contains w ′ explicitly, and since |∂ ℓ r w ′ | r n−ℓ−1 in the vicinity of r = 0 we have the above mentioned regularising effect. The estimates are then routinely performed using the the product rule (A.403) and Lemma 4.14. The outcome is To estimate the last line of (5.346) the crucial insight is that which follows from (5.310) with a = −γ − 1. Therefore the left-hand side above is in fact quadratic in K 1 [θ]. We now estimate the high-order derivatives of the above left-hand side using the product rule (A.403), Lemma 5.6, and Remark 5.7. By analogy to the proof of Lemma 5.9, we obtain On the other hand, when i = N , D N K 1 [θ] contains a top-order term D N +1 H in which case we have to use (C.428) with loss of √ ε: We next discuss the rest of terms in (5.345). Using the product rule (A.403) and (5.341) we have With this decomposition, Lemma 4.14 (applied with b = 0), Lemma 4.8, and the case-by-case analysis analogous to the proof of Lemma 5.13 we obtain the bound where we have used N +2 n = 2( 4 3 − γ) − δ * at the last step. Similarly In order to estimate ετ m P [φapp]H 2 φ 2 app r (the last term on the right-hand side of (5.345)) Using the product rule (A.403) , we may use (5.341) to further decompose it into a linear combination of A 31 HA 32 H where A 31 , A 32 ∈ P i1 , P i2 , i 1 + i 2 = ℓ 3 + 1, i 1 , i 2 ≤ ℓ 3 . We next recall (4.228). Applying (4.227) and the case-by-case analysis analogous to the proof of Lemma 5.13 we obtain the bound Proof of (5.376). Recall that R ε P is defined through (2.82). To estimate the first term on the right-hand side of (2.82) we use the following crude bound By (2.126), Proposition 2.8 we can bound the last line above by To estimate the second term on the right-hand side of (2.82) we first note that (5.380) When we apply D i to the first term on the right-hand side of (5.380) we use the product rule (A.403) to break down the resulting expression into a linear combination of terms of the form with A 1 ∈ P ℓ1 and A 2,3 ∈P ℓ2,ℓ3 with ℓ 1 + ℓ 2 + ℓ 3 = i. By Proposition 2.8 we have Combining the previous line with (4.222) and (4.223) we can bound the absolute value of (5.381) by To bound the last term on the right-hand side of (2.82) we can use the refined expansion (5.378) to obtain the bound where we have used (2.106), (2.104), (2.73), and (2.78), and the bound |R J | τ Mδ . By an analogous argument we have the bound (r∂ r ) ℓ h M τ (M+1)δ . Using the last two bounds, the product rule, Proposition 2.8, by analogy to the above we obtain n , the claim follows from (5.379), (5.382), and (5.383). Proof of (5.376). Since S (φ app ) = rτ −m S(φ app ), from (2.71) and (5.375)-(5.376) we obtain where we have used the bound τ − i−1 n ≤ τ − i n , for τ ∈ (0, 1].
As a corollary, we obtain the following bound for the source terms.
We now integrate the energy identity (4.180) over the time interval [κ, τ ], τ ≤ 1, and obtain by virtue of Proposition 4.7 conclude that there exists a universal constant C 0 > 2 such that The positivity of δ andδ = min{δ * , δ 2 } guarantees that the last time integral on the right-most side of (5.391) is finite and bounded independently of the constant κ. As a consequence of (5.391) and Proposition 4.7 we conclude Since by the local well-posedness theorem Proposition D.1, the map τ → S N κ (τ ) is continuous, a standard continuity argument applied to (5.392) implies that there exist 0 < σ * , ε * < 1 such that for any 0 < σ < σ * the following is true: for any choice of initial data (H, H τ ) τ =κ satisfying and any 0 < ε < ε * the solution exists on the interval [κ, 1] and satisfies the uniform-in-κ bound Justification of the a priori assumptions (4.182) and (5.268). The size restrictions 0 < ε < ε * , 0 < σ < σ * for ε * , σ * sufficiently small are necessary to ensure that the a priori assumptions (4.182) and (5.268) can be consistently recovered from the standard continuity argument. First let (ℓ 1 , ℓ 2 ) = (0, 2). The embedding inequality (C.430) immediately gives for 0 ≤ ℓ 1 + ℓ 2 ≤ 2, (ℓ 1 , ℓ 2 ) = (0, 2). Now for (ℓ 1 , ℓ 2 ) = (0, 2), it suffices to derive the bound for is a classical solution to (3.162), we may use the equation directly: Using (5.394), (4.204), (4.207) it is easy to see that the first three terms of the right-hand side are bounded by τ For the fourth term, by (4.199), |τ 2 c[φ]| τ δ+ 2 n q −γ−1 ( r n τ ) and moreover by (C.424), we have w D2H Hence, the fourth term is also bounded by τ r , we deduce that the second line is also bounded by τ It is now clear that there exists a universal C so that if we choose σ ′ = C(ε * + σ * ) the bound (4.182) is consistent and can be justified by a classical continuity argument. The same comment applies to (5.268).
6 Compactness as κ → 0 and proof of the Theorem 1.6 Let B k be the Hilbert space generated by the norm: From the theory of weighted Sobolev spaces [26,27], we deduce that B k is compactly embedded into B k−1 for k ≥ 1. Let a family of given initial data (H κ 0 , H κ 1 ) satisfy the uniform bound In particular, this gives the uniform bound of (κ Fix such a sequence κ j and initial data H κj 0 and H κj 1 . Now let (H κj , ∂ τ H κj ) be the solution to the initial value problem (3.157) with initial data H κj 0 and H κj 1 given by Theorem 1.11. Consider its well-defined trace at time τ = 1 (i.e. t = 0 in the original coordinates). Since S N κj (τ = 1) < C 0 σ 2 + ε 2M+1 for all κ j , in particular we have the uniform bound where we have used the crude bound ε 2M+1 ≤ ε 2 . Therefore, there exists a subsequence of κ j , denoted by κ j again and We now consider the solution of (3.157) with the final value (H 0 , H 1 ) at time τ = 1. By the local well-posedness theory obtained similarly as in Proposition D.1, there exists a unique solution (H, ∂ τ H) to (3.157) on a maximal interval of existence (T, 1] for some T < 1. We claim that T = 0. To see this, assume the opposite, i.e. 0 < T < 1. Then for each κ j ∈ (0, T 2 ], consider the sequence of solutions (H κj , ∂ τ H κj ) to (3.157) with given initial condition H Claims (1.37)-(1.38) follow easily by going back to the (s, r)-coordinate system, which in turn give (1.39)-(1.41). This completes the proof of Theorem 1.6.
Data at s = 0. Note that the initial conditions (1.25) that correspond to the obtained collapsing solution are now given by In particular, by the smallness of weighted norms of H and Hardy-Sobolev embeddings, we conclude We may now express the initial densityρ 0 and the initial velocity vector fieldũ 0 (at time s = 0) in Eulerian variables. Let Y = χ 0 (r)y = φ(1, r)y,χ 0 (R) = χ 0 (r), R = |Y | = rχ 0 (r). By (1.11)

By (1.24) we haveρ
(6.397) From (1.6) we then conclude , Since by (1.57) w α (r) = 1 − cr n + o r→0 (r n ) for some c > 0 in the vicinity of r = 0, we conclude that we have the expansioñ (6.398) This formula in view of (6.396) gives a quantified sense in which the initial density is flat about the origin.
Proof. If there is no 1 r in a given D i , we are done since it is the same as D i . Suppose 1 r appears in D i . Then we may write D i X =D i−j−1 1 r Y j X for some Y j ∈ P j . Apply Lemma A.1 to get the L 2 bound for D i X by D i−j Y j X. If Y j does not have 1 r , we are done. If it does, then we repeat the same argument by writing D i−j Y j =D i−j−k−1 1 r Z for some Z ∈ P k and applying Lemma A.1 etc. The repetition ends in at most ⌊ i+1 2 ⌋ steps.
Another appealing feature of P i andP i is that they give rise to an algebraic structure via the following Leibniz rule: Lemma A.4 (Product rule). Let i ∈ N be given.
(a) For any A ∈ P i the following identity holds: is true for all i ≤ ℓ. We will show that it is true for i = ℓ + 1. First let ℓ be even. Then A ∈ P ℓ+1 is either A = D r A ′ or A = 1 r A ′ for some A ′ ∈ P ℓ . By induction hypothesis, it suffices to show that D r (B ′ f C ′ g) and 1 r (B ′ f C ′ g) for B ′ ∈ P k ′ and C ′ ∈P ℓ−k ′ can be written as a linear combination of (Bf )(Cg) for some B ∈ P k and C ∈P ℓ+1−k . If k ′ is even, we so that they are in the desirable form. Now let ℓ be odd. Then A = ∂ r A ′ where A ′ ∈ P ℓ . We will show that ∂ r (B ′ f C ′ g) for B ′ ∈ P k ′ and C ′ ∈P ℓ−k ′ can be written into a desirable form. As before, we consider k ′ even, odd separately. If k ′ is even, we write Note that D r B ′ ∈ P k ′ +1 and D r C ′ , 1 r C ′ ∈P ℓ+1−k ′ , and hence it has the desirable form. If k ′ is odd, (A.404) is an easy consequence of (A.403), because any A ∈P i can be written as A = A ′ ∂ r for some A ′ ∈ P i−1 . Write A(f g) = A ′ ((∂ r f )g) + A ′ ((∂ r g)f ) and apply (A.403) with A ′ to obtain the desired result.
Lemma A.5 (Chain rule). Let a ∈ R, i ∈ N be given and fix a vectorfield W ∈P i . Then for any sufficiently smooth f the following identity holds for some real constants c k,i1,...,i k .
Proof. The proof relies on an induction argument. Let i = 1. Then W = ∂ r and ∂ r (f a ) = af a−1 ∂ r f which verifies (A.405). Suppose (A.405) is true for all i ≤ ℓ. Then we will show that it is true for i = ℓ + 1. First let ℓ be even. Then W ∈P ℓ+1 can be written as W = ∂ r W ′ for some W ′ ∈P ℓ . By induction hypothesis, To this end, we first observe because ℓ is odd, there exists at least one index m, 1 ≤ m ≤ k whose corresponding i m is odd. Now let W j = W ′ j for j = m and W m = 1 r w ′ m . Then it is easy to see that all properties are satisfied so that the expression has its desirable form required by (A.405) for i = ℓ + 1.
The next lemma implies that ∂ i r X can be expressed as a linear combination of admissible operations belonging to P i . Note that the other way around is not true in general: for instance, D r can't be expressed in terms of ∂ r only. Hence, our energy built upon D i 's controls ∂ i r 's as well as P i 's. Lemma A.6. Let i ∈ N be given.
where c A i 's are constants.
Proof. The proof is based on the induction on i. Let i = 1. Then we have Since D r and 1 r belong to P 1 , (A.406) holds. Suppose (A.406) is true for all i ≤ ℓ. We will show that (A.406) holds for i = ℓ + 1. By induction hypothesis, Now if ℓ is odd, ∂ r A is an admissible vector field belonging to P ℓ+1 , and hence we are done. If ℓ is even, then we write ∂ r A = D r A − 2 1 r A Then D r A and 1 r A are both admissible vector fields belonging to P ℓ+1 and thus (A.406) holds for i = ℓ + 1.
The same conclusion holds in Lemma A.6 when we replace A ∈ P i by A ∈P i . We also write a few useful identities relating high-order D, r∂ r , and ∂ r derivatives. The same conclusion holds when we replace D j by A ∈ P j .
(iv) For any j ∈ Z >0 there exist constantsĉ k , k ∈ {0, . . . , j} such that The same conclusion holds when we replaceD j by A ∈P j .
Proof. The above statements follow easily by induction.
Next we present the commutator identities useful for derivation of high order equations as well as for high order estimates.
Lemma B.2. Let i ∈ Z >0 be given and let e, X be sufficiently smooth functions. For given differential operators A and B, let Proof. Proof of (B.416). The proof is based on the induction on i. Let i = 1. Then by using the identity D r ∂ r X = ∂ r D r X + 2 r 2 X, D 1 (e∂ r X) = D r (e∂ r X) = eD r ∂ r X + ∂ r e∂ r X = e∂ r D r X + ∂ r eD r X − 2∂ r ( e r )X which yields (B.416) for i = 1. Suppose (B.416) is valid for all i ≤ ℓ. It suffices to show (B.416) for i = ℓ + 1. We will verify it when ℓ is odd. The other case (ℓ is even) will follow similarly. By using the induction assumption, D ℓ+1 (e∂ r X) = ∂ r e∂ r (D ℓ X) + ℓ∂ r eD ℓ X + where A = D 1 = ∂ r + 2 r or A =D 1 = ∂ r .
Proof. The proof is based on the standard Hardy inequality and the cutoff function argument. For A =D 1 , see [33]. The case A = D 1 = ∂ r follows from |D r u| 2 χ 2 r 2 dr, which immediately yields (C.419) with A = D r . The localized Hardy inequality near the boundary follows similarly. We note that w ∼ 1 − r in the vicinity of the boundary r = 1.
As a consequence of the above lemma, we have that for any smooth u : B 1 (0) → R and any m ∈ Z >0 Proof. The proof follows from the Sobolev embedding inequality u ∞ u L 1 + ∂ r u L 1 .
While the summed norms generated by D i orD i or ∂ i r are all equivalent away from the origin, the ordered derivatives near the origin require some attention. We start with (C.423). First, we will verify (C.423) in the case B i =D i , i = 0, 1, . . . , m + 1.   (|u| 2 + |∂ r u| 2 + |∂ 2 r u| 2 )dr +ˆ3 4 0 |D 3 u| 2 r 2 χ 2 dr This finishes the proof of (C.424).
The same argument gives the following bound for ru ∞ : where we apply (C.419) just once near the origin, since ru ∞ ru L 1 + ∂ r (ru) L 1 , to derive the first sum.
We now recall the Hilbert space B N with the norm: In what follows, we will derive the weighted L 2 and L ∞ embedding inequalities for functions in B N based on Lemma C.1, Lemma C.2. . Then w is strictly positive for r ∈ [0, 1 2 ] and hence w α+2k−N w α+k for r ∈ [0, 1 2 ], the first pieces are trivially bounded by the right-hand sides. Now for the second pieces, we apply (C.420) repeatedly (N − k) times starting with a − 2 = α + 2k − N ≥ 0 to deduce the result.
Likewise, for (C.428), we divide the integral into two pieces. Then the integral restricted to [0, 1 2 ] is bounded by E N because w α+2k+1−N w α+k+1 for r ∈ [0, 1 2 ]. For the integral from 1 2 to 1, we first observe that (τ + 2 3 M g ) for r ∈ [ 1 2 , 1] is bounded from below and above by positive constants and hence by applying (C.420) repeatedly (N − k) times, we deduce the desired bound. Proof. The proof follows in the same spirit as in the previous lemma. For w k D k+2 H in (C.431), we first apply (C.423) with u = w k D k+2 H and m = 3 + ⌊α⌋ − k: where we take B i = D i for k even and B i =D i for k odd. Then it is easy to see that the first sum is bounded by our energy since k + 4 ≤ N . For the second sum, by using the product rule and smoothness of w, we first note that |B i D k (w k D k+2 H)| 2 i j=0 w 2k−2j |W j H| 2 where W j ∈ P k+2+i−j . Therefore, by further using Lemma A.3, the second sum is bounded by 4+⌊α⌋−k i=0 i j=0ˆ1 1 4 w α+⌊α⌋+6−2j |D k+2+i−j H| 2 dr But note that w α+⌊α⌋+6−2j = w α+k+2+i−j+(4+⌊α⌋−k−j) w α+k+2+i−j because j ≤ i ≤ 4 + ⌊α⌋ − k. Furthermore, the total number of derivatives appearing is k + 2 + i ≤ 6 + ⌊α⌋ = N . Hence we obtain the desired bound. Other cases in (C.431) and (C.433) follow in the same way. (C.432) and (C.434) can be obtained similarly by applying (C.426) instead of (C.423).
Sketch of the proof of Proposition D.1. The proof of Proposition D.1 follows by the well-posedness proof for the compressible Euler system of Jang & Masmoudi [35,36]. The argument of [35,36] will render the existence theory based on an appropriate approximate scheme and a priori bounds. To apply the result of [35,36], we will design the approximate scheme for H := D r H and H from H. We construct j th approximations (H j , ∂ τ H j ) and (H j , ∂ τ H j ) as follows. The first approximation for j = 1, we use the initial data: let (H 1 , ∂ τ H 1 ) = (D r H κ 0 , D r H κ 1 ) and (H 1 , ∂ τ H 1 ) = (H κ 0 , H κ 1 ) where S N κ (H κ 0 , H κ 1 ) ≤ σ 2 . Inductively we obtain the approximate solutions (j + 1) th approximations as follows: For j ≥ 1, let (H j+1 , ∂ τ H j+1 ) be the solution to the initial value problem for the following linear PDE: Note that the schemes mimic the equation (4.175) for i = 1. The subscript j implies that the coefficients appearing in the expression are evaluated by using H j , ∂ τ H j . With the bounds S N κ (H j , ∂ τ H j ) < ∞ depending only on ε and σ, the existence of (H j+1 , ∂ τ H j+1 ) follows from the duality argument in [35,36]. By defining H j+1 by H j+1 = 1 r 2ˆr 0 H j+1 (r ′ ) 2 dr ′ and based on a priori estimates, we also deduce S N κ (H j+1 , ∂ τ H j+1 ) < ∞ whose bound depends only on ε and σ. As j → ∞, after extracting a subsequence, we obtain the limit (H, ∂ τ H) of (H j , ∂ τ H j ) that solves (3.157) in [κ, T ] for some T = T (σ) > κ with S N κ (H, ∂ τ H) < ∞.